# I A common 2nd order ODE from dynamics but...

#### Tunalover

Consider a simple single degree-of-freedom (SDOF) spring-mass-dashpot dynamic system with spring rate k, mass m, and viscous damping coefficient c. Dimension x is the absolute displacement of the mass. The base input translation is y. A dot notation indicates differentiation with respect to time, t.

A free body diagram of the mass is:

and the base input acceleration is a half-sine pulse of magnitude A and duration T:
$$y=\begin{cases} Asin \frac {\pi t} T, ~0 \le \ t \le \ T \\ 0,~ t \gt T \end{cases}$$
Determine the displacement x as a function of time t:
This system is actually a reasonably realistic simulation of the response of a suspension system from an impact of duration T.
The free body diagram summation of forces provides the following ODE:
$$m \ddot x = k(y-x) + c(\dot y - \dot x)$$
To simplify the problem, make the following substitutions:
Let $z=x-y,$ then $\dot z=\dot x -\dot y$, and $\ddot z = \ddot x - \ddot y$.
After making these substitutions and simplifying,
$$m \ddot z + c \dot z + kz = -m \ddot y$$
To put this in more traditional form,
$$\ddot z + \frac c m \dot z + \frac k m z=- \ddot y$$
The ratios of constants appear often in vibration theory and they are related to the natural frequency $\omega_n$ (rad/sec), and dimensionless critical damping ratio $\xi$ as follows:
$$\omega^2_n = \frac k m$$ and
$$\frac c m = 2 \xi \omega_n$$
then the ODE appears like this after substituting:
$$\ddot z + 2 \xi \omega_n \dot z + \omega^2_n z = -2 \xi \omega_n \ddot y$$
and the final form is:
$$\ddot z + 2 \xi \omega_n \dot z + \omega^2_n z = -2 \xi \omega_n \begin{cases} Asin \frac {\pi t} T, ~0 \le \ t \le \ T \\ 0,~ t \gt T \end{cases}$$
This is as far as I'm taking it. Take this the rest of the way on your own and let me know how it turns out for you. I'm not really looking for the solution but I'm rather putting this out there as a challenge to all mathematically-inclined forum participants. I'll put forward the solution in an attachment a few months from now or to anyone who contacts me asking for it before then. If you find a published solution please do not post to this thread and spoil it for everyone!

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#### Dr.D

This is a routine introductory problem in vibrations. It appears in countless sources and even more college lectures.

#### Tunalover

Could have fooled me. When I took dynamics in college this was never described. Perhaps you could provide a source or two that shows the solution to this problem?

#### Dr.D

It is the sort of problem that is is often discussed in engineering courses on vibrations. I have not looked for a specific example, but probably Timoshenko, Den Hartog, or Steidel's books on Vibrations will have this example.

Mentor

#### berkeman

Mentor
Update -- After a little Moderation, the thread is re-opened.

#### Dr.D

If I had time, I would go locate a citation for you, but I'm going to have to take a pass at this point as I'm rather busy. You are welcome to come dig through my old lecture notes, if you'd like, as I know I have lectured on this problem several times.

#### berkeman

Mentor
Here is a reply sent to me by @bigfooted during the thread closure:
This problem is in the chapter 'Vibration and Time Response' in Meriam and Kraige's "Engineering Mechanics, Volume Two: Dynamics".
The only 'not-so-standard' thing about your ODE is that your forcing function is a Heavyside function. But these problems can be solved easily with the Laplace transform. You can find how to do that in e.g. the section 'Differential Equations with Discontinuous Forcing Functions' in Boyce and DiPrima's 'Elementary Differential Equations and Boundary Value Problems'.

Also, please be kind in your replies :-) You could have remarked instead that the discontinuous forcing function is not very standard.

#### Dr.D

Thank you, bigfooted. Good job.

"A common 2nd order ODE from dynamics but..."

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